Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions

TL;DR

This paper introduces a method using Cauchy's integral formula to accurately compute asymptotic expansions of solutions to turning point problems, exemplified by Bessel functions with complex arguments.

Contribution

It develops a novel approach employing Cauchy's integral formula and exponential Liouville-Green expansions for high-precision computation of asymptotic coefficients.

Findings

High accuracy computation of Airy-type expansions of Bessel functions.

Effective method for asymptotic coefficient calculation in turning point problems.

Application to complex argument Bessel functions demonstrates practical utility.

Abstract

Linear second order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to high order of accuracy. The method employs a certain exponential form of Liouville-Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.